Conjugate direction particle swarm optimization based approach to determine kinetic parameters from differential scanning calorimetric data

Thermochimica Acta 676 (2019) 271–275

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Conjugate direction particle swarm optimization based approach to determine kinetic parameters from differential scanning calorimetric data

T

Hao Wang, Zichao Guo , Wanghua Chen ⁎

Department of Safety Engineering, School of Chemical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, China

ARTICLE INFO

ABSTRACT

Keywords: Optimization Differential scanning calorimetry N-order reaction Autocatalytic reaction Conjugate direction particle swarm optimization

Optimization of the kinetic parameters with thermal analysis data is an essential issue. In this work, the conjugate direction particle swarm optimization (CDPSO) approach, as a global stochastic optimization algorithm that is suitable to high-dimension optimization problem, is first employed to estimate the kinetic parameters with DSC data. This algorithm combines the global optimization ability of the particle swarm optimization (PSO) and the ability to escape from the local extremum by the conjugate direction algorithm (CD) to find the globally optimal solutions. This algorithm does not require estimation of the initial values of the kinetic parameters. The validation of this method is verified by two cases: DCPO decomposition and CHP decomposition. By comparing the experimental and calculated heat flow results, the accuracy of the fitted kinetic parameters is verified. These two cases prove the effectiveness of CDPSO algorithm in the estimation of high-dimension kinetic parameters using DSC data.

1. Introduction The earliest research on analyzing the reaction kinetics by thermal analysis methods can be traced back to the 1920s [1,2]. The reaction mechanism and kinetic parameters obtained can be used to characterize the thermal stability of reactants, assess the reaction hazard, optimize the chemical process and design the emergency relief system [3–7]. Differential scanning calorimeter (DSC), as a pre-screening experimental equipment, has been widely used in thermal safety assessment of chemical process. Compared to other thermal analysis techniques like Accelerating Rate Calorimeter (ARC), DSC presents several advantages, such as small sample size, time saving, and simple operation. The tested curve by DSC reflects the thermal characteristics of the substances. In general, kinetic analysis of DSC results can be divided into two main categories: model free approach and model based approach. The first one usually refers to the iso-conversion method. Thermodynamics Dynamics Committee advised to determine the kinetics of substances by using the iso-conversion method for at least 3–5 different temperature increase rate test results or multiple different temperature isothermal curves [8]. With respect to the model based approach, the first step of kinetics analysis is choosing the kinetic model. Different kinetic models should be considered to analyze different kinds of reaction [9]. After the kinetic model has been properly determined, the next work is to estimate

⁎

the kinetic parameters. Due to the strong non-linear characteristics of the kinetic model, estimation of the kinetic parameters is usually based on nonlinear optimization algorithms [10]. The nonlinear optimization algorithms are usually divided into two categories: the local ones and the global ones. When using the local optimization algorithms, estimation of the initial values of the kinetic parameters must be carried out first [11]. This process is highly important since an improper estimation of the initial values of the kinetic parameters may lead to the wrong kinetic parameters, which indicates that the local optimization algorithms strongly depend on the initial estimated values of kinetic parameters. To deal with this drawback, the global optimization algorithms are recommended to be used. In our previous work [10], one global optimization algorithm, namely particle swarm optimization (PSO) algorithm, was successfully applied to estimate the kinetic parameters using the ARC data. The particle swarm optimization (PSO) algorithm was first proposed by Kennedy [12]. PSO is a global optimization algorithm based on random distribution points. It can find the optimal solution within a given range of values, but the ability to search for local optimal solutions are weak. Although the good results were obtained by pure PSO algorithm, we find that when the number of the required estimated kinetic parameters is high, the results obtained by pure PSO algorithm are prone to fall into the local optimal solution, in other words, the global optimal solution cannot be found. This drawback is in good agreement with the

Corresponding author. E-mail address: [email protected] (Z. Guo).

https://doi.org/10.1016/j.tca.2019.05.009 Received 27 January 2019; Received in revised form 29 April 2019; Accepted 13 May 2019 Available online 14 May 2019 0040-6031/ © 2019 Elsevier B.V. All rights reserved.

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conclusion in reference [13]. It is well accepted that the Conjugate direction (CD) method, a nonderivative optimization method proposed by Powell [14], has a strong ability to optimize local solution for multi-dimensional situation. Therefore, in this article the conjugated direction particle swarm algorithm (CDPSO) will be employed to estimate the kinetic parameters with a high number using DSC data. This article will be constructed as follows: first the CDPSO method will be introduced in detail; then two case studies will be conducted to validate the CDPSO method in estimating the kinetic parameters with a high number. 2. The conjugated direction particle swarm optimization (CDPSO) 2.1. Particle swarm optimization Particle Swarm Optimization (PSO) is a bionic algorithm that mimics the foraging behavior of birds. Each individual bird is called a particle and represents a potential feasible solution. The location of food is the global optimal solution of the objective function. All the particles move in the D-dimensional space and obey a certain rule. Each particle must learn from its best position ( pbest ) and the best position in past iterations in the group ( gbest ). The velocity and the position of the particles is numerically updated following Eqs. (1) and (2). i vti+ 1 = vti + c1 r1 (pbest

xit + 1

=

xti

+

x ti ) + c2 r2 (gbest

x ti )

(1) (2)

vti+ 1

where is the position of particle i at iteration t, is the velocity of i particle i at iteration t, pbest is the best position of particle i in the past iterations, gbest is the best position in the past iterations in the group, x ti+ 1 is the position of particle i at iteration t+1. The position and velocity update process of particles are illustrated in Fig. 1. Once the predefined stop criterions are fulfilled, the iteration process will stop. Though the PSO algorithm has been widely used, this algorithm possibly traps to the local minima and consequently give local optimum results in the high-dimensional cases, which has been proven by Mo [13].

x ti

vti

2.2. Conjugate direction algorithm If the number of kinetic parameters is equal to n, then the kinetic parameter estimation process can be considered as an n-dimensional optimization problem. Each kinetic parameter corresponds to one base vector. The conjugate direction (CD) algorithm is an optimization algorithm that does not require derivation of the objective function. It was proposed by Powell in 1964 [14]. The basic steps of this algorithm

Fig. 2. The flowchart of CDPSO used in this work.

are described as follows: Step 1. A point A is randomly taken as the initial search point, and the n-dimensional base vector are taken as the initial linearly independent directions. Optimize the objective function along one arbitrarily selected direction from point A and produce the optimal point B. Then set the iteration number k to be equal to 0. Step 2. Set point B as the initial search point and optimize the objective function (introduced in Section 3.3) along the n directions one by one by linearly search method. The optimal solution of each direction is then taken as the initial search point for the next direction optimization until all of the n directions have been searched. Finally we get point C. The new direction of BC must be conjugate with the first direction. We carry out linearly search along BC from point C, and then get an optimal point E. After replacing the first direction with BC , we can get new n directions. Let k=k+1. In step 2, (n+1) times linearly search have been conducted for optimization.

Fig. 1. The position and velocity update of particles. 272

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Table 1 Dynamic DSC test conditions of DCPO. Sample mass, mg

Temperature rise rate, K min−1

Temperature range, ℃

Flow rate of protective gas N2, mLmin−1

2.5 ± 0.04

2,4,8,10

35 350

50

Fig. 3. Fitting results (a) with the logarithmic forms and (b) without the logarithmic forms of Ea and A.

Step 3. Return to step 1 and repeat the loop until k=n. Herein point E is reset to be the initial search point. After n times loops, we can get an optimal solution D. The point D point must be closer to the optimal solution than point A.

flow curves of DSC test with CDPSO. The objective function is defined using the least square method, as shown in Eq. (3). m

final

1

start

minf =

2.3. Conjugate direction particle swarm optimization

(qexp

qsim ) 2

(3)

where qexp and qsim are the experimental and simulated DSC heat flow data, respectively. The term of m in Eq. (3) means the different temperature rise rate tests. In this work, all of the kinetic parameter estimation process requires to fit the heat flow data of four different temperature rise rate tests at the same time.

PSO is a random algorithm, and the initial particles are arbitrary. There are more than one initial particles, and each point cooperates with each other. At the beginning of the algorithm, the value of the objective function decreases rapidly. After several iterations, the value of the objective function decreases at a slower speed and eventually may fall into a local extremum, especially for high-dimensional optimization cases. The conjugate direction algorithm starts from a certain initial point, and the solution is better than the initial point. However, it depends on the selection of the initial point. The algorithm can quickly converge when the initial point is well, otherwise it will go through a long time to calculate and sometimes it is impossible to get a good solution. But, it is difficult to get an appropriate initial point by person. Conjugate direction particle swarm optimization (CDPSO) combines the advantages of the particle swarm optimization and the conjugate direction algorithm. The process of CDPSO is shown in Fig. 2. At the configuration stage, about 50 initial particles are randomly generated and the corresponding values of objective function are calculated. The best particle is denoted as gBest. Then it goes into the PSO stage. PSO stops when it falls into a local extremum (generally the gbest does not change over 20 iterations) and returns an optimal solution (gBest,PSO). Set the gBest,PSO as the initial particle of CD and run CD once. In general, CD can jump out of the local extremum value and obtain a better solution (gBest,CD) than gBest,PSO. If the fitting coefficient R2 is higher than 0.9999, gBest,CD will be output as the global optimization result; Or substitute the initial gBest by gBest,CD and continue the optimization loop again until 4 iterations have been done.

3.2. Experiments Differential Scanning Calorimetry (DSC) is commonly used to investigate the thermal decomposition mechanism of chemicals. In this study, the calorimeter employed was DSC1 produced by METTLERTOLEDO. The samples used were Dicumyl peroxide (DCPO) and Cumyl hydroperoxide (CHP). Both DCPO and CHP were purchased from SHANGHAI LINGFENG Chemical Reagent Co., Ltd. and used without more purification. The samples were sealed in high-pressure stainless steel crucibles. DSC tests for each sample were conducted at four different heating rates: 2, 4, 8 and 10 K/min. 3.3. Conversion of particles dimensions In the thermal decomposition kinetics equation, the kinetics parameters can vary in a wide range, for example, the pre-exponential factor A may be in 1010–1030 and activation energy Ea may be in 104˜106 J/ mol. Such wide ranges span over several orders of magnitude, which are unfavorable to the optimization of CDPSO since the forward speed of the particles is relatively slow and the optimal solution may not be obtained within several iterations. Table 2 The optimal kinetic parameters of DCPO determined by CDPSO with the logarithmic forms of Ea and A.

3. Nonlinear fitting and analysis of dynamic DSC heat flow data by CDPSO 3.1. Objective function This work proposes to estimate kinetic parameters by fitting heat 273

Kinetic parameters

A /s−1

Ea /kJ mol−1

n

Optimal solution Lu et al. [16]

2.45 × 1012 ± 3.48 × 1011 5.32 × 1013

123.56 ± 0.49 124.58

0.91 ± 0.004 1

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Table 3 Dynamic DSC test conditions of CHP. Sample mass, mg

Temperature rise rate, K min−1

Temperature range, ℃

Flow rate of protective gas N2, mLmin−1

1.52 ± 0.01

2,4,8,10

50–250

50

Fig. 4. The fitting results of CHP data by (a) CDPSO and (b) pure PSO approaches using BP model.

To deal with this problem, we convert A and Ea into logarithmic form with the natural constant as the base, as showing in Eq. (4).

For such a kind of reaction, the Benito-Perez model (BP model) is generally used to depict the dynamic behaviors. The simulated heat flow (qsim ) calculated based on BP model can be expressed as

(4)

x1 = ln(A), x2 = ln(Ea)

Where A is replaced with x1 and Ea is replaced with x2. Obviously, the numerical searching range can be reduced into an order of magnitude. To show the advantage of logarithmic form of A and Ea, the DSC data of decomposition of dicumyl peroxide (DCPO) was employed. The decomposition of dicumyl peroxide (DCPO) was experimentally confirmed to be a single N-order reaction [15]. The formulation of heat flow for the single N-order reaction can be expressed by

qsim = Q

dX Ea = Qexp ( )(1 dt RT

X )n

qsim = Q

dX = Q A1 e dt

Ea1 RT (1

X )n1 + A2 e

Ea2 RT X n2

(1

X )n3

(6)

where A1 is the pre-exponential factor of the initial reaction and A2 is to the autocatalytic reaction, s−1; Ea1 is the activation energy of the initial reaction and Ea2 is to the autocatalytic reaction, J/mol; n1, n2 , n3 are the reaction order. From Eq. (6), one can find that there are seven kinetic parameters in BP model. In this case, it is hardly possible to select the proper initial values for all of the seven parameters. The sample of CHP with purity of 80 w%-85 w% CHP is purchased from Aladdin Industrial Corporation. The dynamic DSC test conditions are present in Table 3. To compare the CDPSO and the pure PSO, the initial particles in the configuration stage are the same. The fitting results obtained by these two approaches are shown in Fig. 4. The fitted DSC test data and configuration of CDPSO are the same. The fitting coefficient for these two approaches are R2 = 0.9863 and R2 = 0.8698, respectively, which

(5)

where Q is the overall released heat on the whole DSC test, J; X is the fraction conversion of reactant; T is the reaction temperature, K; A is the pre-exponential factor, s-1; Ea is the activation energy, J/mol; n is the reaction order. The value of Q is directly determined by integrating the DSC heat flow profiles. Thus, only the three kinetic parameters, namely Ea, A and n, are needed to be estimated. The dynamic DSC test conditions of DCPO are present in Table 1. After fitting the DSC test data following the CDPSO method, the results obtained with and without the logarithmic forms of Ea and A are shown in Fig. 3. The fitted DSC test data and configuration of CDPSO are the same. The fitting coefficient for these two cases are R2 = 1 and R2 = 0.95697, respectively. Obviously, the fitted results with the logarithmic form are better than that without the logarithmic form. In addition, the optimization with and without the logarithmic form takes 7.26 s and 14.19 s, respectively, which indicates that the fitting process with the logarithmic form are more efficient. The optimal kinetic parameters obtained with the logarithmic form and corresponding parameters obtained by Lu et al. [16] are shown in Table 2. 3.4. Case study 2—Nonlinear fitting of DSC data with Benito-Perez autocatalytic reaction model To show that the CDPSO method is more effective than the pure PSO method, the decomposition of Cumyl hydroperoxide (CHP) is employed. CHP is usually used for ethylene pyrolysis gasoline arsenic removal and polymerization initiator. The thermal decomposition reaction of CHP was experimentally verified to be autocatalytic [17–19].

Fig. 5. Comparison of experimental and calculated conversion degree vs temperature profiles for CHP data fitted by CDPSO. 274

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Table 4 The optimal kinetic parameters of CHP. Kinetic parameters

A1 /s−1

A2 /s−1

Ea1 /kJ mol−1

Ea2 /kJ mol−1

n1

n2

n3

Optimal solution J.-R. Chen et al. [20]

8.52 × 1010 ± 8.07 × 1010 7.42 × 108

2.65 × 103 ± 1.53 × 103 9.60 × 109

117.21 ± 8.42 102.47

43.73 ± 2.06 95.81

2.43 ± 0.75 1.80

1.63 ± 0.06 4.02

0.98 ± 0.02 3.00

apparently indicates that CDPSO could produce more accurate results than pure PSO when the number of kinetic parameters are high. In addition, the comparison of experimental and calculated conversion degree vs temperature profiles for CHP data fitted by CDPSO are shown in Fig. 5. Obviously, the experimental conversion degree profiles are in good agreement with the calculated ones, which offer more powerful evidence to the validation of CDPSO. Moreover, we also compare the calculation time for these cases. They are 21.55 s and 50.65 s, respectively, for the CDPSO and the pure PSO approach. This verifies that the CDPSO approach is more efficient than the pure PSO when encountering the case of high number of kinetic parameters. The optimal kinetic parameters by CDPSO and corresponding results by other researchers are shown in Table 4.

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4. Conclusion In summary, the CDPSO optimization approach has been first employed to estimate the thermal kinetic parameters using DSC data. Compared to the traditional local optimization approaches, CDPSO does not require input of the initial value of kinetic parameters, which make the estimation of the kinetic parameters easier. This CDPSO approach combines the advantages of particle swarm optimization and conjugate direction algorithm, which make this approach effective for high dimensional optimization problem. The thermal decomposition processes of DCPO and CHP conducted by DSC are employed to validate the CDPSO. The results show that this CDPSO can estimate the kinetic parameters more effectively and accurately relative to pure PSO. Acknowledgements This work has been financially supported by National Key R&D Program of China (2017YFC0804701-4) and the Fundamental Research Funds for the Central Universities (30917011312). References [1] J.H. Flynn, Thermal analysis kinetics— Past, present and future, Thermochim. Acta 203 (1992) 519–526.

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